Editing Slope (section)

==Difference of slopes==
[[File:Missing_square_puzzle.svg|thumb|right|200px|The illusion of a paradox of area is dispelled by comparing slopes where blue and red triangles meet.]]
An extension of the idea of angle follows from the difference of slopes. Consider the [[shear mapping]] 
:<math>(u,v) = (x,y) \begin{pmatrix}1 & v \\ 0 & 1 \end{pmatrix}.</math>
Then <math>(1,0)</math> is mapped to <math>(1,v)</math>. The slope of <math>(1,0)</math> is zero and the slope of <math>(1,v)</math> is <math>v</math>. The shear mapping added a slope of <math>v</math>. For two points on <math>\{(1,y):y\in\R\}</math> with slopes <math>m</math> and <math>n</math>, the image
:<math>(1,y)\begin{pmatrix}1 & v \\ 0 & 1\end{pmatrix} = (1, y + v)</math>
has slope increased by <math>v</math>, but the difference <math>n-m</math> of slopes is the same before and after the shear. This invariance of slope differences makes slope an angular [[invariant measure]], on a par with circular angle (invariant under rotation) and hyperbolic angle, with invariance group of [[squeeze mapping]]s.<ref>{{Cite journal|last1=Bolt|first1=Michael|last2=Ferdinands|first2=Timothy|last3=Kavlie|first3=Landon|date=2009|title=The most general planar transformations that map parabolas into parabolas|url=https://projecteuclid.org/euclid.involve/1513799118|journal=Involve: A Journal of Mathematics|language=EN|volume=2|issue=1|pages=79–88|doi=10.2140/involve.2009.2.79|issn=1944-4176|doi-access=free|access-date=2021-05-22|archive-date=2020-06-12|archive-url=https://web.archive.org/web/20200612185227/https://projecteuclid.org/euclid.involve/1513799118|url-status=live}}</ref><ref>{{Wikibooks-inline|Abstract Algebra/Shear and Slope}}</ref>